Physics-Informed Neural Network Methods for Solving Eigenvalue Problems in Neutron Diffusion Equations
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摘要: 为推动物理信息神经网络（PINNs）在堆芯物理计算中的实际应用，并实现深度学习方法与核物理模型的深度融合，提升其在复杂物理系统中的应用潜力，本文提出了一种适用于多种物质排布的多群中子扩散本征值神经网络模型。该模型基于物质区域特征采样设计自适应权重策略，且无需对中子通量密度进行归一化处理。通过对单群多物质算例和两群BIBLIS基准题的求解计算结果表明：两者有效增殖系数绝对误差分别为529.6pcm（1pcm=10−5）和112.5pcm，且各组件功率相对误差均小于5%，初步验证了本文模型的准确性和有效性。本文研究通过物理约束与神经网络模型的有机结合，为复杂堆芯的数值模拟提供了一条新的技术路径，有望促进深度学习方法在反应堆物理设计、安全分析及多物理场耦合计算中的工程化应用。Abstract: To promote the practical application of Physics-Informed Neural Networks (PINNs) in core physics calculations and to achieve a deep integration of deep learning methods with nuclear physics models, thereby enhancing their potential in complex physical systems, this paper proposes a multi-group neutron diffusion eigenvalue neural network model applicable to various material arrangements. In this model, an adaptive weighting strategy is designed based on the characteristic sampling of the material region, and the flux normalization is not required. By solving the single-group multi-material case and the two-group BIBLIS benchmark problem, the calculation results show that the absolute errors of the keff for both cases are 529.6 pcm (1pcm=10−5) and 112.5 pcm, respectively, and the relative error of each assembly's power is less than 5%. These results preliminarily verify the accuracy and effectiveness of this model. This study, through the combination of physical constraints and neural network models, provides a new technical pathway for the numerical simulation of complex reactor cores and is expected to promote the engineering application of deep learning methods in reactor physics design, safety analysis, and multi-physics coupled calculations.
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图 2 多群中子扩散本征值神经网络模型流程图
Figure 2. Flowchart of Multi-group Neutron Diffusion Eigenvalue Neural Network Model
图 4 单群算例归一化中子通量密度计算结果
Figure 4. Normalized Neutron Flux Calculation Results of the Single-group Case
图 5 单群算例归一化功率计算结果
Figure 5. Normalized Power Calculation Results of the Single-group Case
图 6 两群BIBLIS基准题几何模型
1~8—不同材料的编号。
Figure 6. Geometric Configuration of the Two-group BIBLIS Case
图 7 两群BIBLIS算例归一化中子通量密度计算结果
Figure 7. Normalized Neutron Flux Calculation Results of the Two-group BIBLIS Case
图 8 两群BIBLIS基准题归一化功率计算结果
Figure 8. Normalized Power Calculation Results of the Two-group BIBLIS Case
表 1 单群算例各材料截面数据
Table 1. Cross Sectional Data for Each Material in the Single-group Case
材料 Σa /cm−1 Σs /cm−1 vΣf /cm−1 石墨 0.150 0.50 0 燃料 0.075 0.53 0.079 水 0.010 0.89 0 
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表 2 单群算例keff及归一化中子通量密度/归一化功率L2相对误差计算结果
Table 2. Numerical Results of keff and L2 Relative Error of Normalized Neutron Flux and Normalized Power for Single-group Case
$ {k}_{\mathrm{e}\mathrm{f}\mathrm{f}} $ L2相对误差/% 参考解 模型解 绝对误差/pcm 归一化中子
通量密度归一化功率 1.011791 1.006495 529.6 3.031 2.441
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表 3 两群BIBLIS各材料截面数据
Table 3. Cross Sectional Data for Each Material in the Two-group BIBLIS Case
材料 g Dg /cm Σa,g /cm−1 (vΣf)g /cm−1 Σs,1→2 /cm−1 1 1 1.4360 0.009504 0.005871 0.01775 2 0.3635 0.075006 0.096067 0 2 1 1.4366 0.009679 0.006191 0.01762 2 0.3636 0.078436 0.103580 0 3 1 1.3200 0.002656 0 0.02311 2 0.2772 0.071596 0 0 4 1 1.4389 0.010363 0.007453 0.01710 2 0.3638 0.091408 0.132360 0 5 1 1.4381 0.010003 0.006191 0.01729 2 0.3665 0.084828 0.103580 0 6 1 1.4385 0.010132 0.006429 0.01719 2 0.3665 0.087314 0.109110 0 7 1 1.4389 0.010165 0.006191 0.01713 2 0.3679 0.088024 0.103580 0 8 1 1.4393 0.010294 0.006429 0.01703 2 0.3680 0.090510 0.109110 0
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表 4 两群BIBLIS基准题keff及归一化中子通量密度/归一化功率L2相对误差计算结果
Table 4. Numerical Results of keff and L2 Relative Error of Normalized Neutron Flux and Normalized Power for Two-group BIBLIS Case
$ {k}_{\mathrm{e}\mathrm{f}\mathrm{f}} $ L2相对误差/% 参考解 模型解 绝对误差/pcm 快群归一化中子通量密度 热群归一化中子通量密度 归一化功率 1.025103 1.023978 112.5 2.257 2.186 2.258
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